Advanced Algebra S2

Edit 0 24 …

Advanced Algebra Lesson Objectives


Semester 2 Semester 1

Chapter 8
In this chapter you will:
  • use a third variable to write parametric equations that separately define x and y
  • simulate objects in motion with parametric equations
  • review the trigonometric ratios - sine, cosine, and tangent

8.3 The student will

  • Learn about right angle trigonometry.
    • Define the trigonometric ratios sine, cosine, and tangent
    • Use trigonometric ratios to find the unknown lengths of the sides of a right triangle
    • Use inverse trigonometric ratios to find the unknown angle measures of a right triangle
    • Use trigonometric ratios and parametric equations to model and solve problems about motion
    • Find bearings corresponding to angles with the x- and y-axes, and find angles corresponding to given bearings.


Chapter 5
In this chapter you will:
  • write explicit equations for geometric sequences
  • use exponential functions to model real-world growth and decay scenarios
  • review the properties of exponents and the meaning of rational exponents
  • learn how to find the inverse of a function
  • apply logarithms, the inverses of exponential functions

5.1 The students will

  • Learn about exponential functions
    • Write explicit equations for geometric sequences
    • Define exponential function and recognize y=abx as the parent function
    • See real-world growth and decay situations, and recognize that the exponential function models growth when b is greater than 1 and models decay when b is less than 1
    • Learn about half-life and doubling time
    • Evaluate an exponential function using either the explicit equations or graphical methods.


5.2 The students will

  • Learn about properties of exponents and power functions
    • Introduce algebraic proof
    • Review the properties of exponents, specifically, rewriting powers with the same base
    • Introduce the parent power function y = axb, and distinguish it from the exponential function y = abx
    • Find some solutions to power equations using the properties of exponents
    • Introduce rational exponents as a means of solving equations


5.3 The students will

  • Learn about rational exponents and roots
    • Discover that x½ is equivalent to external image 2daa5f53f1bccd91041fe1506cc9f47e.pngDefine rational exponents as equivalent roots or roots raised to powers
    • Introduce the root notation
    • Formally define the point-ratio form of an exponential function
    • Use the point-ratio form to find an exponential equation through two points.


5.4 The students will

  • Learn about applications of exponential and power equations
    • Find solutions to real-world applications of rational-exponential, exponential, and power functions


5.5 The students will

  • Learn about building inverses of functions
    • Define the inverse relation of a function and recognize that the inverse may not be a function or may be a function over a restricted domain
    • Given several points in a function, find inverse points
    • See relation-inverse symmetry across the line y = x
    • Find the composition of functions with their inverses
    • Apply inverses in real-world situations such as time-distance and temperature-pressure


5.6 The students will

  • Learn about logarithmic functions
    • Review solving equations for exponents
    • Define logarithm with base b and common logarithm with base 10
    • Discover, using a calculator, that log(10x) = x
    • Solve logarithmic equations with base 10 and with bases other than 10
    • Establish that the inverse of an exponential function is a logarithmic function


5.7 The students will

  • Learn about properties of logarithms
    • Use the properties of exponents to multiply, divide, and exponentiate with logarithms
    • Learn to change bases so that all logarithms can be calculated using common logarithms
    • Formally define the properties of logarithms


5.8 The students will

  • Learn about applications of logarithms
    • Take the logarithm of both sides of an exponential equation to solve it
    • Use logarithms to straighten a curve so that an algebraic model of the data can be found




Chapter 6
In this chapter you will:
  • use matrices to organize information
  • add, subtract, and multiply matrices
  • solve systems of linear equations with matrices
  • graph two-variable inequalities on a coordinate plane and solve systems of inequalities
  • write and graph inequalities that represent conditions that must be met simultaneously

6.1 The students will
  • Learn about matrix representations
    • Create transition diagrams and transition matrices
    • Relate transition diagrams to matrices
    • Understand matrices as a way to organize information
    • Learn vocabulary associated with matrices: dimension, row, column, entry
    • Find subsequent totals from the given initial value and percentages

6.2 The students will
  • Learn about matrix operations
    • Learn these matrix operation: addition, scalar multiplication, and multiplication
    • Analyze matrix dimensions to determine whether they can be added or multiplied
    • Use matrices as a tool for describing transformations of geometric shapes
    • Understand dilations as stretches or shrinks by the same factor both horizontally and vertically

6.3 The students will
  • Learn about the row reduction method
    • Represent a system of equations as an augmented matrix
    • Interpret reduced row-echelon form
    • Understand and use row operations in a matrix
    • Use matrix row reduction to solve systems of equations

6.4 The students will
  • Learn about solving systems with inverse matrices
    • Find the inverse of a matrix if it exists
    • Use inverse matrices to solve systems of equations
    • Understand the characteristics of an identity matrix

6.5 The students will
  • Learn about systems of linear inequalities
    • Review linear inequalities and the solution of an inequality, including multiplying and dividing by a negative number
    • Write inequalities to describe given real-world constraints
    • Graph systems of inequalities
    • Interpret the meaning of the points within a feasible region
    • Find the vertices of a feasible region

6.6 The students will
  • Learn about linear programming
    • Learn about uses of linear programming and optimization
    • Apply linear programming to situations with two variables
    • Optimize a function over a feasible region


Chapter 7
In this chapter you will:
  • find polynomial functions that fit a set of data
  • study quadratic functions in general form, vertex form, and factored form
  • find roots of a quadratic equation from a graph, by factoring, and by using the quadratic formula
  • define complex numbers and operations with them
  • identify features of the graph of a polynomial function
  • use division and other strategies to find roots of higher-degree polynomials

7.1 The students will
  • Learn about polynomial degree and finite differences
    • Define polynomial, monomial, binomial, and trinomial
    • Determine the degree of a polynomial by using finite differences
    • Explore falling objects
    • Write polynomials in general form
    • Use finite differences and systems of equations to find a polynomial function that fits a data set

7.2 The students will
  • Learn about equivalent quadratic forms
    • Understand the correspondence between the zeros of a polynomial function and the roots of an equation
    • Use the zero-product property to find the roots of equations
    • Comprehend the relationships among the general form, the factored form, and the vertex form of a quadratic equation
    • Relate the vertex form of a quadratic equation to the parent function y=x2

7.3 The students will
  • Learn about completing the square
    • Explore projectile motion
    • Understand completing the square as one way to convert the general form of a quadratic equation to vertex form
    • Use formulas to convert the general form of a quadratic equation to vertex form
    • Use the vertex form of a quadratic equation to solve problems involving maximums or minimums

7.4 The students will
  • Learn about the quadratic formula
    • Use the vertex form of a quadratic equation to find the equation's roots
    • Derive the quadratic formula from the vertex form of a quadratic equation
    • Use the quadratic formula to solve application problems

7.5 The students will
  • Learn about complex numbers
    • Define complex numbers as numbers of the form a + bi, where i2 = -1
    • Identify and find the conjugate of a complex number
    • Find nonreal solutions as conjugate pairs
    • Explore arithmetic computations with complex numbers
    • Introduce a complex plane with a real axis and an imaginary axis

7.6 The students will
  • Learn about factoring polynomials
    • Explore functions defined by 3rd-degree polynomials (cubic functions)
    • Use graphs of polynomial equations to find the roots and write the equation in factored form
    • Relate the graphs of polynomial equations to the number and types of roots

7.7 The students will
  • Learn about higher-degree polynomials
    • Investigate extreme values and the end behavior of polynomial functions
    • Identify possible degrees of a polynomial function by looking at its graph
    • Identify and find the lowest-degree polynomial that has given roots
    • Find additional roots when given one complex root
    • Relate the graphs of polynomial equations to the number and types of roots

7.8 The students will
  • Learn more about finding solutions
    • Learn to find zeros of higher-degree polynomial functions when only a few of the zeros can be found from its graph
    • Use the Rational Root Theorem to identify rational numbers that might be zeros
    • Confirm that a number is a zero of a polynomial function by using the Factor Theorem and use long division to factor and find zeros of higher-degree polynomial functions
    • Understand and use synthetic division as a shortened form of long division for polynomials


Chapter 9
In this chapter you will:
  • use the distance formula to find the distance between two points on a plane and to solve distance and rate problems
  • study rational functions and learn special properties of their graphs
  • add, subtract, multiply, and divide rational expressions

9.1 The students will
  • Learn about using the distance formula
    • Derive the distance formula from the Pythagorean Theorem
    • Apply the distance formula to problems involving a locus of points

9.6 The students will
  • Learn about rational functions
    • Investigate inverse variations
    • Define rational function
    • Graph and find the equations of transformations of the parent function y = 1/x
    • Use rational functions to solve mixture problems and for other applications

9.7 The students will
  • Learn about graphs of rational functions
    • Identify characteristics of the graph of a rational function from its graph
    • Learn to write the equation of a rational function from its graph
    • Rewrite a function as a rational function
    • Review factoring
    • Identify holes, vertical asymptotes, x-intercepts, y-intercepts, and horizontal asymptotes of rational functions

9.8 The students will
  • Learn about operations with rational expressions
    • Learn to add, subtract, multiply, and divide rational expressions
    • Review factoring
    • Simplify rational expressions in order to find x-intercepts, vertical asymptotes, and holes